alcuin and euler: of minuscules and maths

One: Anatomy and taxonomy

Consider them individually, as I, and perhaps you, have always done. One, a minuscule named for Charlemagne’s scribe, the abbot of Tours (735–804), was first drawn 17 years ago by Gudrun Zapf-von Hesse (see Zapf von Hesse, Bindings · Handwritten Books · Typefaces · Examples of Lettering and Drawings, 2002) and issued in digital form in 1991. The other, a super-family of mathematical fonts that comprises Roman, Greek, fraktur, and script forms, was named for the Swiss mathematician, Leonhard Euler (1707–1783) and is the product of a two-year collaboration between Hermann Zapf and Donald Knuth (see Knuth and Zapf, AMS Euler – A New Typeface for Mathematics, reprinted in Chapter 17 of Knuth, Digital Typography, 1999).

Consideration of the two faces together is likely a novel proposition: Although both were conceived in the Carolingian spirit and drawn in the same studio, they have lived in very different worlds from the start. While Alcuin drifts freely throughout the PostScript plenum, AMS (American Mathematical Society) Euler is largely confined to the small sphere of TeX/LaTeX. Yet when their Romans are set side-by-side, a remarkable resemblance of one to the other is immediately apparent; whereas some glyphs are akin to brother and sister, others are more like identical twins in different clothing, or perhaps more appropriately, like husband and wife in matching rompers.

It is only fair to say that, while Alcuin is intended for text use, the Roman of Euler merely comprises a set of math literals: letters not intended for use in text, but rather in mathematical formulas, denoting not word parts but constants, scalars, vectors, and the like. Nonetheless, a study of the two in tandem is meritworthy; in doing so, we may better understand how two highly similar faces – contemporaneously crafted by arguably the most influential espoused typographers of the twentieth century – can serve very different purposes.

There are of course some differences between Alcuin and Euler, and these are described here and are shown in the accompanying figures. The most conspicuous contrast between the lower cases of Alcuin and Euler lies in a fundamental difference of forms in the a and g: two-storey in the former, and single-storey in the latter. Another notable distinction is that the ascenders of Euler are considerably shorter – 85% the height – than those of Alcuin. Other differences are:

1. The serifs are generally sharper in Alcuin than in Euler
2. Serifs at stem bases of letters such as d, h, and i have a greater angle to the horizontal in Euler than in Alcuin
3. In the letter e, the angle of the crossbar to the horizontal is greater in Euler than in Alcuin
4. The serifs at the stem apexes have a greater angle to the horizontal in Alcuin than in Euler
5. The dots over i and j are smaller and are raised higher in Alcuin than in Euler
6. The axis angle from the vertical is greater in Euler than in Alcuin

Salient differences between the typefaces in the upper case are fewer and appear more restricted to the fascia:

1. In Alcuin, the serifs on stem apexes are angled to the horizontal; this is a motif throughout the upper case
2. A motif pervasive in Euler is the presence of foot serifs on stem bases
3. There is a tendency toward unilateral serifs on stem apexes in Alcuin (see, for example, I and J), and toward a mixture of both unilateral (H, K) and bilateral serifs in Euler (I, J, L)

In the creation of Euler, Knuth and Zapf “decided that the new font should have a ‘handwritten’ flavor” (Knuth, Digital Typography, p345). To this end, particular attention was paid to the numerals, and it is perhaps here where we see the biggest differences between the two typefaces. The authors continue: “...there was a chance that Euler’s own handwriting would inspire some feature of the design. And indeed, it turned out that Euler often made the top of the numeral zero pointed instead of round. However, this is a common characteristic of handwriting in general...” It is true: Nowhere in Euler is the appearance of handwritten forms as prevalent as in the numerals.

In order to digitize Euler and prepare it for book and manuscript use, much more work was necessary than was originally foreseen; six years passed from the time Zapf completed his drawings to Euler’s first appearance in print. And most would agree that the result was well worth the wait. But curiously, and perhaps sadly, we rarely see Euler in use. What is it about this typeface that gives authors and publishers pause? Why – relative to other mathematical fonts – is Euler so seldom employed?

Two: Aesthetics and semiotics

“Gentle reader: This is a handbook about TeX, a new typesetting system intended for the creation of beautiful books...” So begins the preface of Donald Knuth’s, The TeXbook, the canonical guide to the use of Knuth’s seminal form of TeX – what is now known as “plain” TeX (Knuth, 1986). The importance of TeX and its variants to the publishing community – particularly, but not limited to, the scientific community – cannot be over-exaggerated. In any bookstore around the world in which you will find mathematical texts – the imprints of Chapman & Hall/CRC, Springer-Verlag, and John Wiley and Sons foremost among them – you will see the fruits of Knuth’s typesetting system, and you’ll feign to deny that these books, regardless of their particular subject matter, are indeed beautiful.

But upon further inspection, you note that these texts – however beautiful in appearance and varied in content – are eerily similar, almost as though a single book designer bore a heavy, unwavering hand in the final formatting; and rather than co-optimizing design and content, he or she stubbornly or perhaps lazily gave a common look to all.

Strange as it may seem, this is more or less just what happened. In the early 1980’s, Leslie Lamport developed a suite of macros for use with TeX, and he called the resulting amalgamation LaTeX. This program's raison d'être is to facilitate even further the formatting of documents, particularly journal articles, reports, and books, enabling authors to attain beauty via simplicity; and LaTeX is now the flavor of TeX that most authors – who happen to serve double duty as book designers – use.

In his guide to the program, Lamport writes, “With LaTeX, you don’t have to worry about formatting while writing your document,” and, “As you are writing your document, you should be concerned with its logical structure, not its visual appearance” (Lamport, LaTeX: A Document Preparation System, 2nd ed, 1994, p7). Though I am a frequent user of the program, as well as an unabashed champion of the user’s guide, these statements have long worried me; the LaTeX approach, one of “logical design,” generally seeks no middle ground – one where aesthetics and content can co-mingle. Rather, the macros of LaTeX operate as though on autopilot; and so we have literally scores of beautiful books that all look pretty much the same.

Further compounding this morphological monotony is a relative homogeneity of letterform: Virtually all documents processed in TeX/LaTeX are done so in its native lingua franca face, Computer Modern. And indeed, there is utility in this uniformity: On a practical level, leaving the defaults in place, not only in TeX but in any word-processing program, facilitates document sharing among users; and in a manner that serves the commonweal more subtly, but perhaps more profoundly, redundancy of font and format increases the transparency of both; over time, the signal-to-noise ratio of the content is maximized.

But what if you want or need to explore the flexibility of type and typesetting within TeX? A particular hobby – or more frankly, a perennial struggle – of mine in mathematical typesetting lies in arranging novel, harmonious cohabitations of Roman and Greek letterforms on the page. Amicable Greco-Roman admixtures exist, of course, but they may quickly lose their allure when one contemplates the extra work required to make them sing with one voice. The existence of Euler, in the absence of any expressly crafted companion face, provides an opportunity to take up the struggle once again. And so what to marry with Euler; how efficiently would this new union communicate mathematics; how aesthetically compatible would this union be; and might this novel combination shift or change the meaning of the mathematics signified therein?

Knuth found a pairing; the debut of Euler was in a book called Concrete Mathematics (1989) that Knuth co-wrote with Ronald Graham and Oren Patashnick, and in which the text face was a Knuth-designed slab-serif called, fittingly, Concrete Roman. With no intention of denigrating Knuth and his colleagues, I’ve always found this to be an odd, immiscible couplet: a slab/typewriter Roman with (mostly) a Carolingian Greek. But not everyone shares my view, of course. At the end of their article on AMS Euler, Knuth and Zapf include an excerpt of an email from University of Chicago statistician Ronald Thisted that addresses both efficiency and aesthetics in Concrete Mathematics: “Incidentally, I find the result of the typography and design to be the most readable technical book I have seen in some time. I am usually fatigued after reading a few pages of most books, but I was able to read all of Chapter 1 without my eyes wandering” (Knuth, Digital Typography, p363).

So it’s a problem of personal aesthetics, then. My own concern – generally, for what is beautiful and harmonious on the printed page, and specifically, for the compatibility of Euler and Concrete Roman in the textbook, Concrete Mathematics – leads me to believe that Euler could be better served. But is that all? At a deeper, and possibly more complex level, does a math book set in Concrete Roman/Euler “say” something different than another whose semantics and syntax are identical but whose letterforms differ? Does the redesigned Smithsonian, which now uses Hoefler Text and Gotham, mean what it did a year ago, when it “spoke” in FB Village and PMN Caecilia? More broadly, when the font changes, does the message remain the same?

(Incidentally, even as I am working through this post, I am typing in 8 pt Verdana with a column width of 3½ inches – thus giving my drafts nearly the same look as the final product rendered in HTML. If instead I were to pound away in 12 pt Times New Roman, it would be akin to doing a dress rehearsal in the wrong costume.)

Well, if the medium is indeed the message, as McLuhan so famously claimed, then the answer is a resounding no! But surely you knew this already. Take, for example, the short, rather direct command – an invitation if you will: “Kiss me!” It is set here in two very different typefaces. If you were to return to your desk after a coffee break to find this phrase – set in Pastonchi (above) – taped to your monitor, you might well be tempted to take your budding office romance to the next level. Set as it is, there is little ambiguity; distilled to its very essence, it says, “I want you now; come over here and kiss me this instant!” Instead of imploring you with Pastonchi, however, suppose your office mate – with some inexplicable intention – decided that Blood of Dracula (below) – in red, no less – might be a nice font selection for this missive. How would you react? You might think, “Kiss you? In dripping blood? Hey, I like you and all, but I’m not ready for Billy Bob and Angelina redux just yet.” In short, the message is confusing: The door to a kiss has been opened, but what lies beyond?

Admittedly, the dichotomy I’ve drawn here is much more extreme than that between Euler and Computer Modern, but the point holds: When the typeface changes, so does the message, and so might the interpretation. Semioticians – scholars who are interested in signs and symbols and their functions – might tell us that, with regard to our example, while the fundamental form (cf. Plato) of the letter and word pattern – or the signifier as linguist Ferdinand de Saussure called it – is the same in both cases (“Kiss me!”), it appears differently; and therefore the concept, labeled the signified, may differ markedly (see top figure; see also Saussure, Course in General Linguistics, 1959, p66; and Helfand, Screen: Essays on Graphic Design, New Media, and Visual Culture, 2001, pp105–110). But what if the context changes? Say that, instead of espying the latter of those two messages during the workday, you receive it in the evening at your Halloween office party. The exquisite timing might just heighten the excitement and arousal, not to mention the cathartic pleasure the inevitable kiss provides.

We need to extend Saussure’s model of the sign (comprising both signifier and signified), then, to envelop it within context (bottom figure), and thereby acknowledge, as is proper, the dependence of both signifier and signified on environment and circumstances. As Eric Gill wrote, “...there is also a norm of letter clothes; or rather there are many norms according as letters are used for this place or purpose or that. Between the occasion wherein the pure sans-serif or mono-line (block) letter is appropriate & that in which nothing is more appropriate than pure fancifulness, there are innumerable occasions” (Gill, An Essay on Typography, 1988, p47). Robert Bringhurst puts it more succinctly: “Typography exists to honor content” (The Elements of Typographic Style, 2002, p17). He also says, “Good typography is like bread: ready to be admired, appraised, and dissected before it is consumed” (p49). But baking good bread requires a certain amount of tinkering and hard work before one gets it right. In order to use AMS Euler, one must obtain the required package and then go about the task of playing matchmaker for it – finding a companion roman and then scaling one or the other to achieve equal x-height. And this, one could suppose, is a reason Euler finds so little use.

A second reason is tied to semiotics and relates to a phenomenon we might call “sign dissonance.” If the relationship between the signifier and the signified is ambiguous (i.e., the Greek beta in Euler may or may not be understood to represent the beta, and from a statistical/mathematical perspective, may therefore be misconstrued as something other than a parameter estimate in a linear regression equation), then readers may not fully understand the text. Fear of sign dissonance – and a desire to promote its opposite – what might be called “sign confluence” – is perhaps another reason why some avoid Euler.

But there are few things as good to eat as freshly-baked bread; and so it is with the use of Euler. In my opinion, it is well worth the extra work and the slight risk in order to produce beautiful text. Near the beginning of this post I wrote that a study of Euler and Alcuin is worthwhile in hopes that we may better understand how two highly similar faces can serve very different purposes. It is also worth our time and energy to scrutinize them in order to find out how they may work together. And indeed, a little experimentation shows that they can work together very well.

I am of the opinion that more rigorous experimentation with Euler, as well as with other fonts that hold promise vis-à-vis mathematics must be prioritized in the coming months and years. The expansion of character sets via OpenType and unicode allows for this, as do recent extensions of TeX and its variants. While a manuscript on 17th century mathematics is nicely set using Computer Modern in the context of an extant LaTeX document class, its content may be honored more justly using a face and a format that, at once, avoid anachronism and promote aesthetic harmony. Simply put, we seem to have eschewed art in favor of technology; can we not add it back into the mix?

Three: Physiology

Allowing Alcuin and Euler to function in concert within the confines of LaTeX requires a bit of work on the part of both, although Alcuin will receive the lion’s share. Because the .fd, .tfm, and .vf (font definition, TeX font metric, and virtual font) files for Alcuin are not publicly available, you will have to create them yourself; here’s how:

1. Make copies of your .afm and .pfb files. If you are working with the light weight of Alcuin, as I am, change the name of your .afm file to one that will work with the program fontinst; the moniker uall8a will do.
2. Run fontinst on uall8a.afm. Doing this will generate a series of .pl and .vpl files, as well as the necessary .fd files.
3. Execute the program pltotf on each of your .pl files, and run vptovf on your .vpl files; this step generates .tfm and .vf files. After doing so, you may delete your .pl and .vpl files.
4. Move the newly-created .tfm and .vf files to the appropriate folders in your fonts directory within your TeX program. Also, move your .fd files to the base folder within your LaTeX directory.
5. Change the name of your .pfb file to uall8a and move it into the type1 folder of your fonts directory.
6. Add the following line to your psfonts.map file, either within dvips (if you will view your documents using GSView), or pdftex (if you plan to use pdfLaTeX): uall8r URWAlcuinT-Ligh "TeXBase1Encoding ReEncodeFont" <8r.enc
7. Call Alcuin in your LaTeX file by adding the following line to your preamble: \renewcommand{\rmdefault}{ual}

Now to add Euler to the mix; fortunately, this is a speedy step:

1. If your LaTeX installation is a relatively recent one, you should have built-in support for Euler in the form of .tfm and .vf files, as well as the necessary font binary (.pfb) files. If you lack these files, you may obtain them from the American Mathematical Society’s website.
2. Obtain the either the euler or the eulervm packages from CTAN and place in your base folder.
3. You will want to scale the x-height of Euler to match that of Alcuin. An inelegant, but effective, way to do this is to change the second entry in the \DeclareMathSizes command, corresponding to the text size you are using. For example, if you specify 12 pt text, you may want to change \DeclareMathSizes{12} {12} {9} {7} to \DeclareMathSizes{12} {11.7} {9} {7}.
4. Call the euler or eulervm packages by adding \usepackage{euler} to your preamble. Refresh your format files, and you should be good to go.
5. If, in testing, you find that the Euler fonts are just a tad light in weight compared to Alcuin, you may wish to call the 7 pt masters, rather than the 10 pt masters.

It may be interesting to note, finally, that Charlemagne’s scribe was also a prominent mathematician; indeed, he is best remembered for his mathematical contributions. Perhaps, then, Euler and Alcuin were meant to be used together.

19-Aug 2003